NUMERICALAND

STATISTICAL METHODS

Prof. Bhupendra T. KesariaAssistant Professor,

SVKM’s Usha Pravin Gandhi College of Management,Vile Parle, Mumbai,Maharashtra, India.

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First Edition : 2018

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PREFACE

Dear Students,

I am extremely happy to come out with this book on “Numerical and Statistical Methods”. Thetopics within the chapters have been arranged in a proper sequence to ensure smooth flow of thesubject. Large number of solved examples are included in all chapters for better understanding ofstudents. I sincerely hope that this book will cater to all your needs in this subject.

I thank my daughter Ekta B. Kesaria for her help in preparing and solving the problems. I alsothank my family for their encouragement and support.

I also thank Mr. Srivastava of Himalaya Publishing House Pvt. Ltd. and his entire staff for theirefforts in publishing this book. We have jointly made every possible effort to eliminate all the errors inthe book, however if you find any, please let us know, because that will help us to improve further.

I am grateful to Dr. Mrs. Anju Kapoor (Principal, Usha Pravin Gandhi College of Management)for her constant encouragement, influence and relentless support.

I thank Prof. Hirendand (Coordinator, B.Sc. I.T., Mulund College of Commerce), Prof. SmrutiNanavaty (Coordinator, M.Sc., I.T., Usha Pravin Gandhi College of Management) and Prof. SwapnaliLotlikar (Coordinator, B.Sc. I.T., Usha Pravin Gandhi College of Management) for their inspirationand support.

I also thank my friends and colleagues for their encouragement and patience.

- Author

CONTENTSNo. Chapter Name Page No.

1 Mathematical Modeling and Engineering Problem Solving1.0 Introduction1.1 Objectives1.2 A Simple Mathematical Model1.3 Conservation Laws

1-6

2 Approximations and Round-off Errors2.0 Introduction2.1 Objectives2.2 Significant Figures (Digits)2.3 Accuracy and Precision2.4 Error Definitions2.5 Round-off Errors2.6 Summary

7-14

3 Trunction Errors and the Taylor Series3.0 Introduction3.1 Objectives3.2 The Taylor Series3.3 Error Propagation3.4 Total Numerical Errors3.5 Formulation Errors and Data Uncertainty3.6 Summary

15-30

4 Solution of Algebraic and Transcendental Equations4.0 Introduction4.1 Objectives4.2 Iterative Methods for Locating Roots

4.2.1 Bisection Method4.2.2 Regula-Falsi Method4.2.3 Newton Raphson Method4.2.4 Secant Method

4.3 Iterative Methods and Convergence Criteria4.3.1 Order of Convergence of Iterative Methods like Bisection and

Regula-Falsi Method4.3.2 Convergence of Newton Raphson Method4.3.3 Rate of Convergence of Secant Method

4.4 Programs and Algorithms4.4.1 Program and Algorithm of Bisection Method4.4.2 Regula-Falsi Method4.4.3 Newton Raphson Method4.4.4 Secant Method

4.5 Summary

31-60

5 Interpolation5.0 Introduction5.1 Objectives5.2 Forward Difference

61-93

5.3 Backward Difference5.4 Newton’s Forward Difference Interpolation5.5 Newton’s Backward Difference Interpolation5.6 Lagrange’s Interpolation5.7 Summary

6 Solution of Simultaneous Algebraic Equation (Linear) using Iterative Methods6.0 Introduction6.1 Objectives6.2 Gauss-Jordan Method6.3 Gauss-Seidel Method6.4 Summary

94-115

7 Numerical Differentiation and Integration7.0 Introduction7.1 Objectives7.2 Numerical Differentiation7.3 Numerical Integration

7.3.1 Trapezoidal Rule7.3.2 Simpson’s 1/3rd Rule7.3.3 Simpson’s 3/8th Rule

7.4 Summary

116-139

8 Numerical Solution of 1st and 2nd Order Differential Equations8.0 Introduction8.1 Objectives8.2 Taylor’s Series8.3 Euler’s Method8.4 Modified Euler’s Method8.5 Runge Kutta Method for 1st and 2nd Order Differential Equations8.6 Summary

140-169

9 Least-Squares Regression9.0 Introduction9.1 Objectives9.2 Linear Regression9.3 Polynomial Regression9.4 Multiple Linear Regression9.5 General Linear Least Squares9.6 Non-linear Regression9.7 Summary

170-187

10 Linear Programming (Graphical Method)10.0 Introduction10.1 Objectives10.2 Meaning of Linear Programming10.3 Basic Requirements10.4 Basic Assumptions10.5 Application of Linear Programming in Business and Industry10.6 Advantages and Disadvantages (Limitations)10.7 Standard Form of Linear Programming Problem

188-221

10.8 Graphical Solution to Linear Programming Models10.9 Special Cases of Linear Programming Problems

10.10 Summary11 Random Variables

11.0 Introduction11.1 Objectives11.2 Discrete Random Variables

11.2.1 Random Variables11.2.2 Discrete Random Variables

11.3 Continuous Random Variables11.4 Probability Density Function11.5 Probability Distribution of Random Variable11.6 Expected Value11.7 Variance11.8 Summary

222-245

12 Distributions12.0 Introduction12.1 Objectives12.2 Discrete Distribution12.3 Continuous Distribution12.4 Normal (or Gaussian) Distribution12.5 Summary

246-274

(1)

STRUCTURE:

1.0 Introduction

1.1 Objectives

1.2 A Mathematical Model

1.3 Conservation Laws

1.0. INTRODUCTION

Knowledge and understanding are prerequisites for the effective implementation of any tool.

This is particularly true when using computers to solve engineering problems. Although they have

great potential utility, computers are practically useless without a fundamental understanding of how

engineering system works.

This understanding is initially gained by empirical means that is by observation and experiment.

However, while such empirically derived information is essential, it is only half the story. Over years

and years of observation and experiments; engineers and scientists have noticed that certain aspects of

their empirical studies occur repeatedly.

The primary objective of this chapter is to introduce you to mathematical modeling and its role

an engineering problem solving. We will also illustrate how numerical methods figure in the process.

1.1 OBJECTIVES

To provide a concrete idea of what numerical methods are and how they relate to engineering

and scientific problem solving.

• Learning how mathematical models can be formulated on the basis of scientific principles

to simulate the behaviour of a simple physical system.

• Understanding how numerical methods afford a means to generalize solutions in a manner

that can be implemented on a digital computer.

• Understanding the different types of conservation laws that lie beneath the models used in

the various engineering disciplines and appreciating the difference between steady-state and

dynamic solutions of these models.

• Learning about the different types of numerical methods we will cover in next section.

MATHEMATICAL MODELING

AND ENGINEERING

PROBLEM SOLVING 1

2 Numerical and Statistical Methods

1.2 A SIMPLE MATHEMATICAL MODEL

• A mathematical model can be broadly defined as a formulation or equation that expresses

the essential features of a physical system or process in mathematical terms.

• Models can be represented by a functional relationship between dependent variables,

independent variables, parameters, and forcing functions.

Dependent Variables = f (independent variables, parameters, forcing functions)

• Dependent variable – a characteristic that usually reflects the behaviour or state of the system.

• Independent variables – dimensions such as time and space, along which the systems

behaviour is being determined.

• Parameters – constants reflective of the systems properties or composition.

• Forcing functions – external influences acting upon the system.

Modeling : Newton’s second Low motion.

• Statement: The time rate of change of momentum of a body is equal to the resultant force

acting on it.

F = ma

⇒ a = m

f ← forcing function

Dependent variable Parameter

(acceleration) (mass of object)

• No independent variable is involved.

• Eg. it can be used to determine the terminal velocity of a free-falling body near the earth’s

surface.

• It describes a natural process or system in mathematical terms.

• It represents an idealization and simplification of reality.

• Ignore negligible details of the natural process and focus on its essential manifestations.

• Exclude the effects of “relativity” that are minimal importance when applied to object and forces

that interact on or about the earth’s surface at velocities and on scales visible to humans.

• It yields reproducible results and can be used for predictive purposes.

• Have generalization capabilities.

Bungee – jumping

• For a body falling within the vicinity of the earth, the net force is composed of two

opposing forces.

F = fD + fu

• The downward Pull of gravity FD.

- The force due to gravity can be formulated as:

FD = mg

- g is the acceleration due to gravity (9.81 m/s2)

Mathematical Modeling and Engineering Problem Solving 3

• The upward force of air resistance fu

- A good approximations is to formulate it as:

Fu = – CdV2

- V is the velocity; Cd is the lumped drag coefficient, accounting for the properties of the

falling object like shape or surface roughness.

- >>The greater the fall velocity, the greater the upward force due to air resistance

Upward force due to air resistance

Downward force due to gravity

• The net force therefore is the difference between and upward force, we can have a

differential equation regarding the velocity of the object.

2v

m

cdg

dt

dv −=

• The exact solution of v cannot be obtained using simple algebraic manipulation but rather

using more advanced calculus techniques (when v(t) = 0, t = 0)

v(t) =

t

m

gedh

Cd

gm tan

Here t is independent variable, v(t) is dependent variable, Cd and m are parameters.

g is forcing function

tan h (x) = xx

xx

ee

ee−

−

+−

Example 1:

• A bungee jumper with a mass of 68.1 kg leaps from a stationary hot air balloon (the drag

coefficient is 0.25 kg/m).

- Compute the velocity for the first 12s of free fall.

Determine the terminal velocity that will attained for an infinite long cord.

V(t) =

t

m

gCdh

Cd

gm tan

4 Numerical and Statistical Methods

mg = CdV2

V = Cd

gm

V(t) =

1.68

)25.0( 8.9 tan

25.0

)1.68( 8.9 th = 51.6938 tan h (0.18977 t)

∴ V(12) = 50.6715

V(100) ≈ 50.6938

−10

−10

−5

−5

0

0

5

5

sin ( )h θcos ( )h θ

tan ( )h θ

10

10

Example 2:

• Using a computer (or a calculator), the model can be used to generate a graphical

representation of the system.

20

v 1 m

/s

4

t1 S

40

60

08

Terminal velocity

12

Example of Numerical Modeling

• Numerical methods are those in which the mathematical problem is reformulated so it can

be solved by arithmetic operations.

E.g., the time rate of change of velocity mentioned earlier:

mg = cdV2

V= Cd

gm

Mathematical Modeling and Engineering Problem Solving 5

ii

ii

tt

tvtv

t

v

dt

dv

−−

=∆∆≈

+

+

1

1 )()( (a finite-difference approximation of the derivate at time ti)

Notice that t

v

dt

dv

t ∆∆=

→∆ 0lim

v t( )i

ti

v t( )i+1

t i + 1

∆v

∆t

0

True slope

t

. • Substituting the finite difference into the differential equation gives,

2v

m

Cdg

dt

dv −=

⇒ 2

1

1 )()()(

iii

ii tvm

cdg

tt

tvtv−=−

−

+

+

⇒ Solve for

step

)(

)(

oldnew

)()( 12

1 iiiii tt

xSlopem

tvCdg

tvtv −

⋅−

+=

+= ++(ti + 1 – ti)

This approach is formally called Euler’s method.

Applying Euler’s method in 2s intervals yields

20

v1 m

/s

4

t1 S

40

60

08

Terminal velocity

Exact analytical solution

Approximatenumericalsolution

12

Approximate slope

ii

ii

tt

tvtv

t

v

−−

=∆∆

+

+

1

1 )()(

6 Numerical and Statistical Methods

How do we improve the solution?

- Smaller steps.

1.3 CONSERVATION LAWS

Conservation laws form the basis of a variety of complicated and powerful model and are

conceptually easy to understand.

Conservation laws provide the foundation for many model functions.

They boil down to

Change = increase – decreases

Can be used to predict changes with respect to time by given it a special name “the time-

variant (or transient)” computation

If no change occurs, the increases and decreases must be in balance.

Change = 0 = increases – decreases

• It is given a special name, the “steady-state” calculation

Example : Fluid flow

Pipe 1

flow in = 100

Pipe 2

flow in = 80

Pipe 3

flow out = 120

Pipe 4

flow out = ?

Fig : A flow balance for steady incompressible fluid flow at the junction of pipes.

For steady-state incompressible fluid flow in pipes flow in = Flow out

• The flow out of the fourth pipe must be 60.

Table 1: Devices and types of balances that are commonly used in the four major areas of

engineering.

For each case, the conservation law upon which the balance is based is specified.